The spin-orbit (\(SO\) ) interaction correction to the scalar propagation constant, denoted \(\delta \beta ^{2(SO)}_{(l,m)}\) , is of the order of \(l\Delta /a^2\)  [1]}, [2]}, [3]}. These corrections are of opposite sign for the spin-orbit aligned case (e.g., \(+l, S=+1\) ) and spin-orbit anti-aligned case (e.g.,\(-l, S=+1\) ), where \(S\)  denotes the spin of the OAM mode, with \(S=\pm 1\)  corresponding to left/right circular polarization. Consequently, we expect these differences to manifest as unequal diagonal elements in the \(2~x~2\)  matrix considered in Section 3.1 and Appendix A. Therefore, to maintain the validity of our results,
we must ensure that these differences
due to the SO effects are much smaller than the splitting of the degenerate spatial modes, \(2\gamma _{l,m}\)  due to ellipticity \(\epsilon \)  (Eq. 22), i.e., the \((l,-l)\)  degeneracy is broken primarily through the ellipticity effects. By examining the analytic form for \(\gamma _{l,m}\)  ( Eq. 22), we estimate the condition under which this may hold.
